Subprime residential mortgage loan securitization and its associated risks have been a major topic of discussion since the onset of the subprime mortgage crisis (SMC) in 2007. In this regard, the thesis addresses the issues of subprime residential mortgage loan (RML) securitization in discrete-, continuous-and discontinuous-time and their connections with the SMC. In this regard, the main issues to be addressed are discussed in Chapters 2, 3 and 4. In Chapter 2, we investigate the risk allocation choices of an investing bank (IB) that has to decide between risky securitized subprime RMLs and riskless Treasuries. This issue is discussed in a discrete-time framework with IB being considered to be regret- and risk-averse before and during the SMC, respectively. We conclude that if IB takes regret into account it will be exposed to higher risk when the difference between the expected returns on securitized subprime RMLs and Treasuries is small. However, there is low risk exposure when this difference is high. Furthermore, we assess how regret can influence IB's view - as a swap protection buyer - of the rate of return on credit default swaps (CDSs), as measured by the premium based on default swap spreads. We find that before the SMC, regret increases IB's willingness to pay lower premiums for CDSs when its securitized RML portfolio is considered to be safe. On the other hand, both risk- and regret-averse IBs pay the same CDS premium when their securitized RML portfolio is considered to be risky. Chapter 3 solves a stochastic optimal credit default insurance problem in continuous-time that has the cash outflow rate for satisfying depositor obligations, the investment in securitized loans and credit default insurance as controls. As far as the latter is concerned, we compute the credit default swap premium and accrued premium by considering the credit rating of the securitized mortgage loans. In Chapter 4, we consider a problem of IB investment in subprime residential mortgage-backed securities (RMBSs) and Treasuries in discontinuous-time. In order to accomplish this, we develop a Levy process-based model of jump diffusion-type for IB's investment in subprime RMBSs and Treasuries. This model incorporates subprime RMBS losses which can be associated with credit risk. Furthermore, we use variance to measure such risk, and assume that the risk is bounded by a certain constraint. We are now able to set-up a mean-variance optimization problem for IB's investment which determines the optimal proportion of funds that needs to be invested in subprime RMBSs and Treasuries subject to credit risk measured by the variance of IE's investment. In the sequel, we also consider a mean swaps-at-risk (SaR) optimization problem for IB's investment which determines the optimal portfolio which consists of subprime RMBSs and Treasuries subject to the protection by CDSs required against the possible losses. In this regard, we define SaR as indicative to IB on how much protection from swap protection seller it must have in order to cover the losses that might occur from credit events. Moreover, SaR is expressed in terms of Value-at-Risk (VaR). Finally, Chapter 5 provides an analysis of discrete-, continuous- and discontinuous-time models for subprime RML securitization discussed in the aforementioned chapters and their connections with the SMC. The work presented in this thesis is based on 7 peer-reviewed international journal articles (see [25], [44], [45], [46], [47], [48] and [55]), 4 peer-reviewed chapters in books (see [42], [50j, [51J and [52]) and 2 peer-reviewed conference proceedings papers (see [11] and [12]). Moreover, the article [49] is currently being prepared for submission to an lSI accredited journal. / Thesis (Ph.D. (Applied Mathematics))--North-West University, Potchefstroom Campus, 2010.

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.

Η παρούσα διδακτορική διατριβή εστίασε το ενδιαφέρον της στην διερεύνηση των αναλογικών ολοκληρωμένων φίλτρων της κατηγορίας ELIN εξωτερικά γραμμικά, εσωτερικά μη-γραμμικά. Συγκεκριμένα μελετήθηκαν και σχεδιάστηκαν, νέες δομές φίλτρων συμπίεσης-αποσυμπίεσης του προς επεξεργασία σήματος στο πεδίο του υπερβολικού ημιτόνου χαμηλής τάσης τροφοδοσίας. Η γοργή ανάπτυξη της μικροηλεκτρονικής στην υλοποίηση συστημάτων υψηλής αξιοπιστίας και απόδοσης μικρού βάρους και όγκου όπως φορητών ηλεκτρονικών πολυμέσων, επικοινωνιών, βιοϊατρικών συσκευών, ωθεί στην σχεδίαση των ολοκληρωμένων κυκλωμάτων που τα απαρτίζουν με μειωμένη κατανάλωση ισχύος και κατ’ επέκταση χαμηλής τάσης τροφοδοσίας. Σ’ ένα ολοκληρωμένο κύκλωμα η ενσωμάτωση αναλογικών και ψηφιακών κυκλωμάτων σε περιβάλλον χαμηλής τάσης τροφοδοσίας, ώστε το κόστος των διατάξεων να διατηρείται χαμηλό, επηρεάζει άμεσα την απόδοση του αναλογικού τμήματος, προτάσσοντας την ανάγκη για νέες αρχιτεκτονικές σχεδίασης του. Οι διατάξεις των αναλογικών φίλτρων αποτελούν συχνά δομικό στοιχείο των ολοκληρωμένων κυκλωμάτων και η διερεύνηση τους για την επίτευξη μεγάλης δυναμικής περιοχής, ηλεκτρονική ρύθμισης της απόκρισης συχνότητας και ταυτόχρονα χαμηλής κατανάλωσης ισχύος, έχει απασχολήσει αρκετά το ενδιαφέρον της ερευνητικής κοινότητας. Προτείνεται αρχικά η συστηματική σχεδίαση φίλτρων υψηλής τάξης στο πεδίο του υπερβολικού ημιτόνου, αποσκοπώντας στη βελτίωση της διαδικασίας υλοποίησης τους, με χρήση ήδη υπαρχουσών μη-γραμμικών διαγωγών υπερβολικού ημιτόνου, συνημίτονου. Η συμπίεση-αποσυμπίεση του προς επεξεργασία σήματος επιτυγχάνεται με την κατάλληλη τοποθέτηση συμπληρωματικών τελεστών ενώ ταυτόχρονα διατηρείται γραμμική η συνολική συμπεριφορά των διατάξεων. Με γνώμονα την εφαρμογή τους σε βιοϊατρικές συσκευές, εξομοιώνεται με δύο διαφορετικούς τρόπους φίλτρο 3ης τάξης leapfrog με συχνότητα αποκοπής 10Hz και τα αποτελέσματα που προκύπτουν συγκρίνονται με αντίστοιχο γραμμικής συμπεριφοράς. Στην συνέχεια, υλοποιείται η σχεδίαση BiCMOS φίλτρων οποιασδήποτε τάξης, δομημένα από μη-γραμμικούς διαγωγούς υπερβολικού ημιτόνου με δυνατότητα λειτουργίας σε υψηλές συχνότητες και σε χαμηλή τάση τροφοδοσίας. Επιλέγονται οι μέθοδοι του γραμμικού μετασχηματισμού, λειτουργικής και τοπολογικής εξομοίωσης ενός πρωτότυπου ελλειπτικού βαθυπερατού 3ης τάξης με συχνότητα αποκοπής 0.5 kHz και κυμάτωση στην ζώνη διέλευσης 0.5dB. Ακολουθεί ανάλυση και σύγκριση των αποτελεσμάτων που προκύπτουν στο περιβάλλον Analog Virtuoso της Cadence Software, της τεχνολογίας AMS CMOS S35 0.35μm, μεταξύ των φίλτρων στο πεδίο του υπερβολικού ημιτόνου καθώς και με αντίστοιχες διατάξεις στο πεδίο του λογαρίθμου τάξης-ΑΒ και (OTA)-C στο γραμμικό χώρο. Προτείνεται ακόμη, η BiCMOS σχεδίαση γενικευμένων φίλτρων 2ης τάξης απλής εισόδου-πολλαπλών εξόδων και πολλαπλής εισόδου-απλής εξόδου, με δυνατότητα λειτουργίας σε περιβάλλον χαμηλής τάσης τροφοδοσίας, ηλεκτρονικής ρύθμισης της συχνότητας αποκοπής ω0 του συντελεστή ποιότητας Q και της ορθογώνιας μεταβολής μεταξύ των. Από την εξομοίωση τους πρόκυψε ότι είναι ενεργειακά αποδοτικότερη η υλοποίηση τους συγκρινόμενα με αντίστοιχα στο πεδίο του λογαρίθμου. Τέλος, στα πλαίσια υλοποίησης βιοϊατρικών εφαρμογών προτείνεται η σχεδίαση ενός μη-γραμμικού τελεστή ενέργειας στο πεδίο του υπερβολικού ημιτόνου για την ανίχνευση αιχμών δραστηριότητας σε νευρωνικά δίκτυα με τάση τροφοδοσίας 0.5V. Ο μη-γραμμικός διαγωγός υπερβολικού ημιτόνου, του οποίου ο διαγραμμικός βρόγχος αποτελείται από (pMOS) τρανζίστορ δομεί τους διαφοριστές και πολλαπλασιαστές τεσσάρων τεταρτημορίων τάξης-ΑΒ που απαρτίζουν την διάταξη. Εξομοιώνεται, κάνοντας χρήση CMOS τρανζίστορ της τεχνολογίας TSMC 130 nm, στο περιβάλλον Analog Virtuoso της Cadence Software και συγκρινόμενος με ήδη υπάρχουσες διατάξεις, παρουσιάζει την μικρότερη κατανάλωση ισχύος. / This present Ph.D dissertation is focused its interest on the design of low voltage analog integrated circuits. Companding (compressing-expanding) systems are Externally Linear, Internally Non-linear (ELIN) processors with potential for low-voltage operation capability. In this direction novel topologies of companding filters in the Sinh-Domain are introduced. Τhe radical technological developments of microelectronics in the systems implementation with high reliability and performance, such as portable electronic devices for multimedia, communications and biomedical systems, demand the design of integrated circuits with reduced power consumption and thus low voltage supply. Integration of analog and digital circuits of systems-on-chip – (SoC) in order to be kept the low cost, directly affects the performance of the analog section, pointing forward the need for novel architectural design. One of the basic building blocks for circuit design is analog filters and their investigation in order to achieved large dynamic range, electronic tuning capability of their frequency characteristics which has gained a significant research effort toward these goals. At first, a new systematic method for designing Sinh-Domain filters is introduced. The proposed method offers the benefits of facilitating the design procedure of high-order Sinh-Domain filters using already introduced building blocks and of the absence of any restriction concerning the type and/or the order of the realized filter function. This is achieved by employing an appropriate set of complementary operators, in order to transpose the conventional functional block diagram representation of each linear operation to the corresponding one into the Sinh-Domain. In order to demonstrate the validity of the proposed systematic method, 3rd order leapfrog low-pass filter with a cut-off frequency at 10Hz which is typical for biomedical applications, has been realized following two alternative approaches and, also, a comparison has been performed among them and a conventional linear filter where the most important performance factors have been taken into account. Continuing, novel BiCMOS sinh-domain filter topologies, derived according to operational emulation, component substitution techniques and Linear Transformation of the corresponding 3rd-order passive prototype filters, with a cut-off frequency at 0.5 kHz and pass-band ripple 0.5dB, is proposed. This has been achieved by utilizing BiCMOS nonlinear ransconductor cells into the sinh domain. An attractive benefit of the proposed filter topologies is their capability for operating in high frequencies and a low-voltage environment. The performance of a leapfrog sinh-domain filter has been compared, using the Analog Design Environment of the Cadence software using AMS CMOS S35 0.35μm, with those of the corresponding log-domain and operational ransconductance amplifier (OTA)-C filters. Furthermore, the design of a family of Sinh-Domain universal biquad filters are introduced offering the benefits of low-voltage operation and, simultaneously, power efficient realizations in comparison with the corresponding already proposed biquads. Also, they have the capability for orthogonal adjustment between the resonance frequency and Q factor of the filter and these parameters could be also electronically adjusted through appropriate dc currents. Thus, they could be considered as attractive blocks for realizing high-performance analog signal processing systems. Finally, a Sinh-Domain topology for realizing the Non Linear Energy Operator (NEO) is introduced. For this purpose, a novel Sinh-Domain differentiator is proposed which offers the benefits of ultra low-voltage operation capability and electronic tuning of the realized time-constant. The whole system is also constructed from a four-quadrant current multiplier, realized by employing appropriately configured non-linear transconductors. Considering a single power supply voltage of 0.5V, the behavior of the proposed Sinh-Domain NEO realization has been simulated using the Analog Design Environment of the Cadence software using TSMC 130 nm design hit.

Les problèmes abordés et les résultats obtenus dans cette thèse se divisent en deux parties. La première concerne l'étude de la valeur asymptotique de jeux dépendant de la fréquence (jeux-FD). Nous introduisons un jeu différentiel associé au jeu-FD dont la valeur se ramène à une équation de Hamilton-Jacobi-Bellman-lsaacs. En affrontant un problème d'irrégularité à l'origine, nous prouvons l’existence de la valeur du jeu différentiel sur [0.1 ] et ceci nous permet de prouver que la valeur du jeu FD converge vers la valeur du jeu continu qui débute à l'état initial 0. Dans la deuxième partie, l'objectif fondamental est la décomposition de l'espace des jeux finis en sous espaces des jeux adéquats et plus faciles à étudier vu que leurs équilibres sont distingués. Cette partie est divisée en deux chapitres. Dans le premier chapitre, nous établissons une décomposition canonique de tout jeu arbitraire fini en trois composantes et nous caractérisons les équilibres approximatifs d'un jeu donné par les équilibres uniformément mixtes et en stratégies dominantes lesquels apparaissent sur ses composantes. Dans le deuxième chapitre, nous introduisons sur l'espace des jeux finis une famille de produits scalaires et nous définissons la classe des jeux harmoniques relativement au produit scalaire choisi dans cette famille. Inspiré par la décomposition de Helmholtz-Hodge appliquée aux jeux par Candogan et al. (2011), nous établissons une décomposition orthogonale de l'espace des jeux finis, par rapport au produit scalaire choisi, en les sous espaces des jeux potentiels, des jeux harmoniques et des jeux non­stratégiques c nous généralisons les résultats de Candogan et al. (2011). / The problems addressed and results obtained in this thesis are divided in two parts. The first part concerns the study of the asymptotic value of frequency-dependent games (FD-games). We introduce a differential game associated to the FD-game whose value leads to a Hamilton-Jacob-Bellman-lsaacs equation. Although an irregularity occurs at the origin, we prove existence of the value in the differential game played over [0.1 ], which allows to prove that the value of the FD-game, as the number of stages tend to infinity, converges to the value of the continuous-time game with initial state 0. ln the second part, the objective is the decomposition of the space of finite games in subspaces of suitable games which admit disguised equilibria and more tractable analysis. This part is divided in two chapters. In the first chapter, we establish a canonical decomposition of an arbitrary game into three components and we characterize the approximate equilibria of a given game in terms of the uniform equilibrium and the equilibrium in dominant strategies that appear in its components. In the second part, we introduce a family of inner products in the space of finite games and we define the class of harmonic games relatively to the chosen inner product. Inspired of the Helmholtz-Hodge decomposition applied to games by Candogan et al (2011 ), we establish an orthogonal decomposition of the space of finite games with respect to the chosen inner product, in the subspaces of potential harmonic and non-strategic games and we further generalize several results of Candogan et al (2011).

Dans le cadre des recherches de la qualité des ressources en eau, l’étude du processus de transfert de masse du sol vers la nappe phréatique constitue un élément primordial pour la compréhension de la pollution de cette dernière. En effet, les éléments polluants solubles à la surface (produits liés aux activités humaines tels engrais, pesticides...) peuvent transiter vers la nappe à travers le milieu poreux qu’est le sol. Ce scénario de transfert de pollution repose sur deux phénomènes : la pluie qui génère la masse d’eau à la surface et la dispersion de celle-ci à travers le milieu poreux. La dispersion de masse dans un milieu poreux naturel comme le sol forme un sujet de recherche vaste et difficile aussi bien au plan expérimental que théorique. Sa modélisation constitue une préoccupation du laboratoire EMMAH, en particulier dans le cadre du projet Sol Virtuel dans lequel un modèle de transfert (modèle PASTIS) a été développé. Le couplage de ce modèle de transfert avec en entrée un modèle décrivant la dynamique aléatoire de la pluie est un des objectifs de la présente thèse. Ce travail de thèse aborde cet objectif en s’appuyant d’une part sur des résultats d’observations expérimentaux et d’autre part sur de la modélisation inspirée par l’analyse des données d’observation. La première partie du travail est consacrée à l’élaboration d’un modèle stochastique de pluie. Le choix et la nature du modèle sont basés sur les caractéristiques obtenus à partir de l’analyse de données de hauteur de pluie recueillies sur 40 ans (1968-2008) sur le Centre de Recherche de l’INRA d’Avignon. Pour cela, la représentation cumulée des précipitations sera assimilée à une marche aléatoire dans laquelle les sauts et les temps d’attente entre les sauts sont respectivement les amplitudes et les durées aléatoires entre deux occurrences d’événements de pluie. Ainsi, la loi de probabilité des sauts (loi log-normale) et celle des temps d’attente entre les sauts (loi alpha-stable) sont obtenus en analysant les lois de probabilité des amplitudes et des occurrences des événements de pluie. Nous montrons alors que ce modèle de marche aléatoire tend vers un mouvement brownien géométrique subordonné en temps (quand les pas d’espace et de temps de la marche tendent simultanément vers zéro tout en gardant un rapport constant) dont la loi de densité de probabilité est régie par une équation de Fokker Planck fractionnaire (FFPE). Deux approches sont ensuite utilisées pour la mise en œuvre du modèle. La première approche est de type stochastique et repose sur le lien existant entre le processus stochastique issu de l’équation différentielle d’Itô et la FFPE. La deuxième approche utilise une résolution numérique directe par discrétisation de la FFPE. Conformément à l’objectif principal de la thèse, la seconde partie du travail est consacrée à l’analyse de la contribution de la pluie aux fluctuations de la nappe phréatique. Cette analyse est faite sur la base de deux relevés simultanées d’observations de hauteurs de pluie et de la nappe phréatique sur 14 mois (février 2005-mars 2006). Une étude statistique des liens entre les signaux de pluie et de fluctuations de la nappe est menée comme suit : Les données de variations de hauteur de nappe sont analysées et traitées pour isoler les fluctuations cohérentes avec les événements de pluie. Par ailleurs, afin de tenir compte de la dispersion de masse dans le sol, le transport de la masse d’eau pluviale dans le sol sera modélisé par un code de calcul de transfert (modèle PASTIS) auquel nous appliquons en entrée les données de hauteurs de pluie mesurées. Les résultats du modèle permettent entre autre d’estimer l’état hydrique du sol à une profondeur donnée (ici fixée à 1.6m). Une étude de la corrélation entre cet état hydrique et les fluctuations de la nappe sera ensuite effectuée en complément à celle décrite ci-dessus pour illustrer la possibilité de modéliser l’impact de la pluie sur les fluctuations de la nappe / Within the research quality of water resources, the study of the process of mass transfer from soil to groundwater is a key element for understanding the pollution of the latter. Indeed, soluble contaminants to the surface (related to human activities such fertilizers, pesticides products ...) can transit to the web through the porous medium that is the ground. This scenario transfer pollution based on two phenomena: the rain that generates the body of water to the dispersion and the surface thereof through the porous medium. The dispersion of mass in a natural porous medium such as soil forms a subject of extensive research and difficult both experimental and theoretical grounds. Its modeling is a concern EMMAH laboratory, particularly in the context of Virtual Sol project in which a transfer model (PASTIS model) was developed. The coupling of this transfer model with input a model describing the dynamics of random rain is one of the objectives of this thesis. This thesis addresses this goal by relying in part on the results of experimental observations and also on modeling inspired by the analysis of observational data. The first part of the work is devoted to the development of a stochastic model of rain. The choice and nature of the model are based on the features obtained from the analysis of data collected rainfall over 40 years (1968-2008) on the Research Centre INRA Avignon. For this, the cumulative rainfall representation will be treated as a random walk in which the jumps and waiting times between jumps are the amplitudes and durations between two random occurrences of rain events. Thus, the probability jumps (log-normal distribution) and that of waiting between jumps (Law alpha-stable) time is obtained by analyzing the laws of probability amplitudes and occurrences of rain events. We show that the random walk model tends towards a subordinate in time geometric Brownian motion (when space step and time step walking simultaneously tend to zero while maintaining a constant ratio), the law of probability density is governed by a Fokker Planck fractional (FFPE). Two approaches are then used to implement the model. The first approach is based on stochastic type and the relationship between the stochastic process derived from the differential equation of Itô and FFPE. The second approach uses a direct numerical solution by discretization of the FFPE. Accordance with the main objective of the thesis, the second part of the work is devoted to the analysis of the contribution of rain to fluctuations in groundwater. We approach this analysis on the basis of two simultaneous records of observations of rainfall amounts and groundwater over 14 months (February 2005-March 2006). A statistical study of the relationship between the signals of rain and fluctuating water will be conducted. Data sheet height variations are analyzed and processed to isolate coherent fluctuations with rain events. In addition, to take account of the mass dispersion in the soil, the mass transport of storm water in the soil layer is modeled by a calculation code transfer (PASTIS model) which we apply input data measured heights of rain. The model results allow between another estimate soil water status at a given depth (here set at 1.6m). A study of the correlation between the water status and fluctuating water will then be performed in addition to that described above to illustrate the ability to model the impact of rain on the water table fluctuations

This work analyzes tuning capabilities of different fully integrated active filter topologies. Work only deals with continuous time active filters. Topologies described in this work differ in type of active element and in method of frequency tuning. Techniques of tunning are proved on second order low pass filter. Filter topologies are compared from tunning capabilities and from point of total harmonic distortion. The main building block of all filters is integrator.

This master's thesis deals with queueing theory and its application in designing node models in packet-switched network. There are described general principles of designing queueing theory models and its mathematical background. Further simulator of packet delay in network was created. This application implements two described models - M/M/1 and M/G/1. Application can be used for simulating network nodes and obtaining basic network characteristics like packet delay or packet loss. Next, lab exercise was created, in that exercise students familiarize themselves with basic concepts of queueing theory and examine both analytical and simulation approach to solving queueing systems.

In dieser Arbeit betrachten wir das Strömungs-Diffusions-(Reaktions)-Problem für passive Markerteilchen, die in zweidimensionalen laminaren Strömungsmustern mit geringem thermischem Rauschen gelöst sind. Der deterministische Fluss umfasst Zellen in Form von Quadraten oder Katzenaugen. In ihnen tritt Rotationsbewegung auf. Einige der Strömungen bestehen aus wellenförmigen Bereichen mit gerader Vorwärtsbewegung. Alle Systeme sind entweder periodisch oder durch Wände begrenzt. Eine untersuchte Familie von Strömungen interpoliert kontinuierlich zwischen Reihen von Wirbeln und Scherflüssen. Wir analysieren zahlreiche numerische Simulationen, die bisherige theoretische Vorhersagen bestätigen und neue Phänomene offenbaren. Ohne Rauschen sind die Teilchen in einzelnen Bestandteilen des Flusses für immer gefangen. Durch Hinzufügen von schwachem thermischen Rauschen wird die normale Diffusion für lange Zeiten stark verstärkt und führt zu verschiedenen Diffusionsarten für mittlere Zeiten. Mit Continuous-Time-Random-Walk-Modellen leiten wir analytische Ausdrücke in Übereinstimmung mit den numerischen Ergebnissen her, die je nach Parametern, Anfangsbedingungen und Alterungszeiten von subdiffusiver bis superballistischer anomaler Diffusion für mittlere Zeiten reichen. Wir sehen deutlich, dass einige der früheren Vorhersagen nur für Teilchen gelten, die an der Separatrix des Flusses starten - der einzige Fall, der in der Vergangenheit ausführlich betrachtet wurde - und dass das System zu vollkommen anderem Verhalten in anderen Situationen führen kann, einschließlich einem Schwingenden beim Start im Zentrum einesWirbels nach einer gewissen Alterungszeit. Darüber hinaus enthüllen die Simulationen, dass Teilchenreaktionen dort häufiger auftreten, wo sich die Geschwindigkeit der Strömung stark ändert, was dazu führt, dass langsame Teilchen von schnelleren getroffen werden, die ihnen folgen. Die umfangreichen numerischen Simulationen, die für diese Arbeit durchgeführt wurden, mussten jetzt durchgeführt werden, da wir die Rechenleistung dafür besitzen. / In this thesis, we consider the advection-diffusion-(reaction) problem for passive tracer particles suspended in two-dimensional laminar flow patterns with small thermal noise. The deterministic flow comprises cells in the shape of either squares or cat’s eyes. Rotational motion occurs inside them. Some of the flows consist of sinusoidal regions of straight forward motion. All systems are either periodic or are bounded by walls. One examined family of flows continuously interpolates between arrays of eddies and shear flows. We analyse extensive numerical simulations, which confirm previous theoretical predictions as well as reveal new phenomena. Without noise, particles are trapped forever in single building blocks of the flow. Adding small thermal noise, leads to largely enhanced normal diffusion for long times and several kinds of diffusion for intermediate times. Using continuous time random walk models, we derive analytical expressions in accordance with numerical results, ranging from subdiffusive to superballistic anomalous diffusion for intermediate times depending on parameters, initial conditions and aging time. We clearly see, that some of the previous predictions are only true for particles starting at the separatrix of the flow - the only case considered in depth in the past - and that the system might show a vastly different behavior in other situations, including an oscillatory one, when starting in the center of an eddy after a certain aging time. Furthermore, simulations reveal that particle reactions occur more frequently at positions where the velocity of the flow changes the most, resulting in slow particles being hit by faster ones following them. The extensive numerical simulations performed for this thesis had to be done now that we have the computational means to do so. Machines are powerful tools in order to gain a deeper and more detailed insight into the dynamics of many complicated dynamical and stochastic systems.

The master’s thesis solves models of queueing systems, which use the property of Markov chains. The queueing system is a system, where the objects enter into this system in random moments and require the service. This thesis solves specifically such models of queueing systems, in which the intervals between the objects incomings and service time have exponential distribution. In the theoretical part of the master’s thesis I deal with the topics stochastic process, queueing theory, classification of models and description of the models having Markovian property. In the practical part I describe realization and function of the program, which solves simulation of chosen model M/M/m. At the end I compare results which were calculated in analytic way and by simulation of the model M/M/m.

Anomale Diffusion ist ein weitverbreiteter Transportmechanismus, welcher für gewöhnlich mit ensemble-basierten Methoden experimentell untersucht wird. Motiviert durch den Fortschritt in der Einzelteilchenverfolgung, wo typischerweise Zeitmittelwerte bestimmt werden, entsteht die Frage nach der Ergodizität. Stimmen ensemble-gemittelte Größen und zeitgemittelte Größen überein, und wenn nicht, wie unterscheiden sie sich? In dieser Arbeit studieren wir verschiedene stochastische Modelle für anomale Diffusion bezüglich ihres ergodischen oder nicht-ergodischen Verhaltens hinsichtlich der mittleren quadratischen Verschiebung. Wir beginnen unsere Untersuchung mit integrierter Brownscher Bewegung, welche von großer Bedeutung für alle Systeme mit Impulsdiffusion ist. Für diesen Prozess stellen wir die ensemble-gemittelte quadratische Verschiebung und die zeitgemittelte quadratische Verschiebung gegenüber und charakterisieren insbesondere die Zufälligkeit letzterer. Im zweiten Teil bilden wir integrierte Brownsche Bewegung auf andere Modelle ab, um einen tieferen Einblick in den Ursprung des nicht-ergodischen Verhaltens zu bekommen. Dabei werden wir auf einen verallgemeinerten Lévy-Lauf geführt. Dieser offenbart interessante Phänomene, welche in der Literatur noch nicht beobachtet worden sind. Schließlich führen wir eine neue Größe für die Analyse anomaler Diffusionsprozesse ein, die Verteilung der verallgemeinerten Diffusivitäten, welche über die mittlere quadratische Verschiebung hinausgeht, und analysieren mit dieser ein oft verwendetes Modell der anomalen Diffusion, den subdiffusiven zeitkontinuierlichen Zufallslauf. / Anomalous diffusion is a widespread transport mechanism, which is usually experimentally investigated by ensemble-based methods. Motivated by the progress in single-particle tracking, where time averages are typically determined, the question of ergodicity arises. Do ensemble-averaged quantities and time-averaged quantities coincide, and if not, in what way do they differ? In this thesis, we study different stochastic models for anomalous diffusion with respect to their ergodic or nonergodic behavior concerning the mean-squared displacement. We start our study with integrated Brownian motion, which is of high importance for all systems showing momentum diffusion. For this process, we contrast the ensemble-averaged squared displacement with the time-averaged squared displacement and, in particular, characterize the randomness of the latter. In the second part, we map integrated Brownian motion to other models in order to get a deeper insight into the origin of the nonergodic behavior. In doing so, we are led to a generalized Lévy walk. The latter reveals interesting phenomena, which have never been observed in the literature before. Finally, we introduce a new tool for analyzing anomalous diffusion processes, the distribution of generalized diffusivities, which goes beyond the mean-squared displacement, and we analyze with this tool an often used model of anomalous diffusion, the subdiffusive continuous time random walk.